Remark 4.1 
Since the mapping \((\omega ,x)\mapsto E^{u}(\omega ,x)\)  is measurable, we can find a measurable basis \(\lbrace e_i{(\omega ,x)}\rbrace _{i=1}^{\dim E^{u}}\)  spanning \(E^{u}(\omega ,x)\)  (see [1]}).
Moreover, for each \(v=(v_i)_{i=1}^{\dim E^{u}}\in \mathbb {R}^{\dim E^{u}}\)  be fixed, define
\(v_{\omega ,x}=:\sum _{i=1}^{\dim E^{u}} v_i e_i{(\omega ,x)}.\) 
